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Optical properties of Solids
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Claudia Ambrosch-DraxlInstitute for Theoretical Physics
University [email protected]
Optical Properties of Solidswithin WIEN2k
Outline
Opti
cs in W
IEN
2k
light scatteringdielectric tensor in the RPAsumrulessymmetrythe band gap problem
Basics
Program
Examples
Outlook
program flowinputs
Raman scatteringbeyond RPA
outputsconvergenceresults
Outline
Opti
cs in W
IEN
2k
light scatteringdielectric tensor in the RPAsumrulessymmetrythe band gap problem
Basics
Program
Examples
Outlook
program flowinputs
Raman scatteringbeyond RPA
outputsconvergenceresults
Exci
ted S
tate
s
Properties & Applications
Dielectric functionOptical absorptionOptical gapExciton binding energyPhotoemission spectraCore level spectraRaman scatteringCompton scatteringPositron annihilationNMR spectraElectron spectroscopy understand physics
characterize materialstailor special properties
Light emitting diodesLasersSolar cellsDisplaysComputer screensSmart windowsLight bulbsCDs & DVDs
Exci
ted S
tate
s
Wavefunction vs. Density
Hartree-Fock:
DFT:
ionization energies
Lagrange parameters
auxiliary functions
Janak's theorem
Koopman's theorem
Opti
cal Pro
pert
ies
Light – Matter InteractionResponse to external electric field E
Linear approximation: susceptibility
conductivity
dielectric tensor
Fourier transform:
Polarizability:
Light Scattering
ES
Opti
cal Pro
pert
ies
intraband transitioninterband transition
Ener
gy
wave vector
EF
band structure
kc
kv
Opti
cal Pro
pert
ies
The Dielectric TensorFree electrons: Lindhard formula
Bloch electrons:
interbandintraband
Interband contribution:
independent particle approximation, random phase approximation (RPA)
Optical "Constants"
Opti
cal Pro
pert
ies
Kramers-Kronig relationsComplex dielectric tensor:
Optical conductivity:
Loss function:
Absorption coefficient:
Reflectivity:
Complex refractive index:
Intraband Contributions
Meta
ls
Drude-like termsDielectric Tensor:
Optical conductivity:
Plasma frequency:
Sumrules
Opti
cal
Pro
pert
ies
Symmetry
Die
lect
ric
Tenso
r
cubic
monoclinic (,=90°) orthorhombic
tetragonal, hexagonal
triclinic
without magnetic field, spin-orbit coupling:
Magneto-optics
Exam
ple
: N
i
cubic
tetragonalwith magnetic field ‖z, spin-orbit coupling:
KK
KK
KK
Exci
ted S
tate
Pro
pert
iesOpen Questions
Approximations used:
Local Density Approximation (LDA)Generalized Gradient Approximation (GGA)
Ground state:
Excited state:Interpretation within one-particle pictureInterpretation of excited states in terms of ground state propertiesElectron-hole interaction ignored (RPA)
Where do possible errors come from?How to treat excited states ab initio?
The Band Gap Problem
many-body perturbation theory: GW approachshift of conduction bands: scissors operator
Electro-affinity
Ionization energy
Band gap
Outline
Opti
cs in W
IEN
2k
light scatteringdielectric tensor in the RPAsumrulessymmetrythe band gap problem
Basics
Program
Examples
Outlook
program flowinputs
Raman scatteringbeyond RPA
outputsconvergenceresults
Program Flow
Opti
cs in W
IEN
2k
SCF cycle converged potential
kgen dense mesh
eigenstateslapw1
Fermi distributionlapw2
momentum matrix elementsoptic
dielectrix tensor components joint
Re Im optical coefficients broadeningscissors operator
kram
"optic"
Inputs
2000 1 number of k-points, first k-point -5.0 2.2 Emin, Emax: energy window for matrix
elements1 number of cases (see choices below)1 Re <x><x>OFF unsymmetrized matrix elements written to
file?
al.inop
ni.inop (magento-optics)800 1 number of k-points, first k-point -5.0 5.0 Emin, Emax: energy window for matrix elements3 number of cases (see choices below)1 Re <x><x>3 Re <z><z>7 Im <x><y>OFF
Choices:1......Re <x><x>2......Re <y><y>3......Re <z><z>4......Re <x><y>5......Re <x><z>6......Re <y><z>7......Im <x><y>8......Im <x><z>9......Im <y><z>
"joint"
Inputs
al.injoint
1 18 lower and upper band index0.000 0.001 1.000 Emin, dE, Emax [Ry] eV output units eV / Ry 4 switch 1 number of columns to be considered0.1 0.2 broadening for Drude model choose gamma for each case!
SWITCH
0...JOINT DOS for each band combination 1...JOINT DOS sum over all band combinations 2...DOS for each band 3...DOS sum over all bands 4...Im(EPSILON) 5...Im(EPSILON) for each band combination 6...INTRABAND contributions 7...INTRABAND contributions including band analysis
"kram"
Inputs
al.inkram
0.1 broadening gamma0.0 energy shift (scissors operator)1 add intraband contributions 1/012.6 plasma frequency0.2 gamma(s) for intraband part
0 1 2 3 4 5 6-20
-10
0
10
20
30
40
50
60
70
80
Re
Im
=0.05eV
Silicon
Die
lect
ric
fun
ctio
n
Energy [eV]
si.inkram
0.05 broadening gamma1.00 energy shift (scissors operator)0....
as number of columsas number of colums
Outline
Opti
cs in W
IEN
2k
light scatteringdielectric tensor in the RPAsumrulessymmetrythe band gap problem
Basics
Program
Examples
Outlook
program flowinputs
outputsconvergenceresults
Raman scatteringbeyond RPA
Outp
uts
Convergence
Exam
ple
: A
l
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.00
25
50
75
100
125
150
175
0 1000 2000 3000 4000 500012.012.112.212.312.412.512.612.712.8
p
k-points in IBZ
165k 286k 560k 1240k 2456k 3645k 4735k
In
terb
an
d Im
Energy [eV]
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
165 k-points 4735 k-points Experiment
Nef
f [el
ectr
ons]
Energy [eV]
Sumrules
Exam
ple
: A
l
Loss Function
Exam
ple
: A
l
0 5 10 15 200
20
40
60
80
100
120
total
intraband
interband
L
oss
fun
ctio
n
Energy [eV]
Outline
Opti
cs in W
IEN
2k
light scatteringdielectric tensor in the RPAsumrulessymmetrythe band gap problem
Basics
Program
Examples
Outlook
program flowinputs
Raman scatteringbeyond RPA
outputsconvergenceresults
0 100 200 300 400 500 6000
20
40
60
80
100
Raman shift [cm-1]
= 35K Ba,Cu = 18K
(zz)
Spe
ctra
l den
sity
[10-7
sr-1]
Theory
Raman Intensities
YB
a2C
u3O
7:
A1
g M
odes
CAD, H. Auer, R. Kouba, E. Ya. Sherman, P. Knoll, M. Mayer, Phys. Rev. B 65, 064501 (2002).
Experiment
0 100 200 300 400 500 6000
20
40
60
80
100
Raman shift [cm-1]
A
1g
Current Developments
Gradient Corrections (GGA)LDA + UExact Exchange (EXX)
Self-interaction correction (SIC)Non-local exchange / screened exchange
Kohn-Sham theory
Generalized Kohn-Sham theory
Time dependent DFT
band gap problemexcitonic effects
non-local effectscorrelation effectsband gap problem
Many-body perturbation theory
GW + Bethe-Salpeter equation
response to time-dependet perturbation
The Bethe–Salpeter Equationeffective Schrödinger equation for the electron-hole pair
Beyond R
PA
Thank you for your attention!
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